Theorem prsspwg 3754

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)

Assertion

Ref	Expression
prsspwg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))

Proof of Theorem prsspwg

Step	Hyp	Ref	Expression
1		prssg 3751	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶))
2		elpwg 3585	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐶 ↔ 𝐴 ⊆ 𝐶))
3		elpwg 3585	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ 𝒫 𝐶 ↔ 𝐵 ⊆ 𝐶))
4	2, 3	bi2anan9 606	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))
5	1, 4	bitr3d 190	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2148   ⊆ wss 3131  𝒫 cpw 3577  {cpr 3595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601

This theorem is referenced by: (None)